19182
J. Phys. Chem. 1996, 100, 19182-19186
Stirring Effect on the Bistability of the Belousov-Zhabotinsky Reaction in a CSTR P. Strizhak† and M. Menzinger* Department of Chemistry, UniVersity of Toronto, Toronto, Ontario M5S 3H6, Canada ReceiVed: August 15, 1996X
Experiments show that the hysteresis of the bistable Belousov-Zhabotinsky reaction contracts along the directions of both the control parameter and the response variable as the stirring rate is decreased. A onedimensional stochastic model reproduces qualitatively and quantitatively the effect of stirring on both steady states and hysteresis limits. The model suggests a classification of known stirring effects. A system behaves effectively one-dimensionally when its hysteresis contracts at low stirring. Otherwise, the system is described by at least two dynamical variables.
1. Introduction The incomplete mixing of reactant feedstreams into a CSTR gives rise to reactor inhomogeneity, which is the main source of stirring effects in nonlinear chemical reactions.1,2 In bistable systems, experiments have shown that the values of steady state concentrations, of stability limits (bifurcation points), and of concentration fluctuations may depend sensitively on stirring.2-6 Theoretical analyses confirmed certain qualitative aspects of these experimental observations.7-12 The stirring dependence of bistability was studied most thoroughly in the chlorite-iodide reaction, both by experiments2-6 and by theory.7,8 Decreased stirring, i.e. increased inhomogeneity, contracts the hysteresis loop by narrowing the parameter range of bistability and by destabilizing both steady states, which tend to approach each other. Qualitatively similar results are found in the Belousov-Zhabotinsky (BZ) reaction with gallic acid13 and in the classical Ce-catalyzed BZ reaction studied here. This inhomogeneity-induced shrinking of the hysteresis, which we classify as the stirring effect of the first type, is however not generic. A second type of stirring effect occurs in the minimal bromate oscillator,14,15 where the hysteresis region expands and the reactive (thermodynamic) branch is stabilized in response to increased inhomogeneity. The objectives of this paper, the first in a series on quantitative interpretations of stirring effects, are to introduce a onedimensional stochastic model that describes stirring effects of the first type and to test its validity by comparing it with experiments in the BZ reaction. The model provides a qualitative and a quantitative interpretation of the observed stirring effects as well as the possibility for their classification. In other words, as far as its bistability is concerned, the BZ system behaves effectively one-dimensionally. Our analysis is also in qualitative accord with results on the clorite-iodide reaction. The generality of the model allows one to suggest that the shrinking of the hysteresis region is the simplest and the most common effect of stirring on bistable chemical systems. The stirring effect of the second type,14,15 on the other hand, cannot be explained by this model and requires one to consider a dynamics with more than one effective variable. The dynamics in a CSTR is governed by the rates of inflow, mixing, and chemical processes. The hydrodynamics is controlled by stirring and by inflow, and the chemistry by the rate † Permanent address: L. V. Pizarzhevskii Institute of Physical Chemistry, Ukrainian Academy of Sciences, prosp. Nauki 31, Kiev, Ukraine, 252039. * E-mail:
[email protected]. X Abstract published in AdVance ACS Abstracts, November 1, 1996.
S0022-3654(96)02504-X CCC: $12.00
law and also by the inflow: the inflow affects both chemistry and hydrodynamics. However, in studying stirring effectssthe effects of hydrodynamics on the chemical reactionsit is preferable to use purely chemical or hydrodynamic control parameters, rather than the flow rate. Most previous experiments have used for experimental convenience the flow rate as the control parameter. We chose instead the concentration of bromide ions in the inflow as the control parameter. 2. Experiments The experiments were conducted in a cylindrical plexiglass CSTR (diameter 31 mm, volume V ) 28 mL; thermostated at T ) 36 °C).5,14 The rectangular stirrer (8 × 15 mm, steel coated by Teflon) was positioned 30 mm above the bottom of the reactor. The stirring rate was varied between 0 < S < 1500 rpm. The state of the system was monitored by a Pt electrode (0.1 mm Pt wire fused in glass) relative to a Hg/HgSO4 reference electrode. The impedance-matched electrode signal was fed via an A/D converter board into a 486-type personal computer. The reactants were peristaltically pumped into the reactor through two ports, located on the opposite sides of the reactor just below the stirrer, at a constant rate 1.99 mL/min, corresponding to the residence time τ0 ) 282 s. Reagents, prepared from analytical grade chemicals, were stored in three different solutions: (1) 8.1 × 10-3 M NaBrO3; 4.35 × 10-3 M Ce2(SO4)3, and (2) 0.03 M malonic acid; (3) the third solution contained the sodium bromide. The bromide concentration was used as the control parameter. Each solution contained 1.5 M H2SO4. Feedstreams containing sodium bromide and solution containing cerium(III) sulfate and malonic acid were premixed before entering the reactor. Experimental conditions were similar to those described elsewhere.16 The hysteresis was mapped out as a function of the bromide concentration in the inflow at different stirring rates. The fluctuating signal was analyzed using the standard statistical approach. We calculated the first and second moments and the probability distribution function. 3. Experimental Results Figure 1 shows the hysteresis at different values of the stirring rate (S). The system response is calculated as the average value of the Pt potential, i.e. as the first moment of the fluctuating signal. At low values of [Br-]0, the system is in the steady state SS I, characterized by high Ce(IV) concentration and autocatalysis turned on. Beyond the upper critical bromide concentration CI, the system switches to the second steady state © 1996 American Chemical Society
Stirring Effect on the Belousov-Zhabotinsky Reaction
Figure 1. Dependence of the system response (first moment of the Pt electrode signal) on the control parameter at different stirring rate (rpm).
SS II, characterized by low Ce(IV) concentration. If [Br-]0 is scanned in the opposite direction, transition from SS II to SS I occurs at the lower hysteresis limit CII. Stirring affects both steady states. Decreasing the stirring rate S shifts SS I toward SS II; that is, it destabilizes this branch, but does not lead to a detectable change of the transition point CI. The stirring effect is more pronounced on the lower branch. As the stirring rate is decreased, the latter shifts up toward SS I and the position of the critical point CII shifts at the same time to higher values of the control parameter. Consequently, decreased stirring contracts the hysteresis in both directions. Stirring affects not only the average value of the potential but also the standard deviation σ2 that characterizes the intensity of fluctuations in the system. The fluctuations depend on the control parameter as shown in Figure 2. They always increase near the transition points. At any [Br-]0, σ2 increases on both branches with decreasing S. Indeed, Figure 3 shows that σ2 is inversely proportional to the stirring rate. This illustrates that the second moment of the signal is proportional to the mixing time, where τmix ≈ S-1. Fluctuations always increase with decreasing stirring rate. The proportionality between σ2 and mixing time confirms experimentally the validity of the coalescence-dispersion model17 to bistable nonlinear systems, for which this proportionality has been established separately.10 The data presented in Figures 1 and 2 illustrate the effect of stirring on the first and second moments, respectively. A more general description is based on the probability distribution function, i.e. the probability of finding the system in a state characterized by a certain value of the Pt electrode potential. Figure 4 shows the probability distribution functions, constructed from the fluctuating signals, for both steady states at two values of the stirring rate. They are all Gaussian. As S decreases, the probability distribution functions become broader and move toward each other. This is the stochastic representation of the fact that the hysteresis shrinks with decreasing stirring.
J. Phys. Chem., Vol. 100, No. 49, 1996 19183
Figure 2. Dependence of the noise (second moment of the Pt electrode signal) on the control parameter at different stirring rate (rpm).
Figure 3. Dependence of the noise σ2 on the inverse stirring rate. [Br-]0 ) 1 × 10-6 M for SS I. [Br-]0 ) 1.5 × 10-4 M for SS II.
4. Stochastic Model of Stirring Effect The Gaussian form of the probability distribution functions and the growth of fluctuations with decreasing stirring allow one to suggest a simple theoretical description of the stirring effect, in which the fluctuations are modeled as a white noise process. We present the simplest form of stochastic description that does not yet attempt to relate the mixing-induced fluctuations to hydrodynamic and chemical factors. A more concrete
19184 J. Phys. Chem., Vol. 100, No. 49, 1996
Strizhak and Menzinger
Figure 4. Probability distribution function of the electrode potential at two different stirring rates (rpm) for both branches, [Br-]0 ) 1.5 × 10-4 M.
y ) x + ξ(t)
(2)
where ξ(t) corresponds to the fast processes of stochastic mixing, and x is a slow variable that is related to the stochastic steady states. The main assumption of our analysis is that the fluctuations are small, i.e. ξ(t) , x(t). This allows one to make a Taylor expansion after substituting eq 2 into eq 1 and to retain only the linear terms. This results in the Langevin equation
x˘ ) f(x) + f′(x) ξ(t)
(3)
We assume that ξ(t) is a white noise process, 〈ξ(t)〉 ) 0, 〈ξ(t) ξ(t′)〉 ) 2Dδ(t - t′), where D is the intensity of fluctuations, which is proportional to the standard deviation Figure 5. Schematic presentation of rate function f(x) which gives bistability.
description of the fluctuations will be given in a forthcoming paper. The main features of our analysis agree with the results of Horsthemke and Hannon.10,11 The simplest case of hysteresis occurs in dynamical systems with a single variable, described by
dy/dt ) f(y,µ)
(1)
where y represents the concentration, and µ a control parameter. Equation 1 describes the bistability in a deterministic system if the rate function f(y,µ) is S-shaped, as is schematically shown in Figure 5. To obtain a simple stochastic model from eq 1, we represent the mixing process as an external additive noise process that perturbs the dynamical variable y, as it evolves according to eq 1.18 To analyze it, we follow the standard procedure described elsewhere.19,20 The source of fluctuations is the imperfect mixing in a CSTR. The fluctuating concentration, which actually represents spatial inhomogeneities, is described by a stochastic variable y(t) that fluctuates due to the mixing of reactor subvolumes with different concentrations, as described by cellular mixing models.17 Stirring tends to average the concentrations due to the mixing of subvolumes. This allows one to represent the stochastic variable y, whose evolution is described by eq 1, as
D ) τσ2
(4)
where τ is a characteristic time scale. The deterministic eq 1 may be considered a limiting case of eq 3 for vanishing fluctuations, i.e. D f 0. Note that the fluctuating term in eq 3 that describes the mixing process is a function of the stochastic variable. Hence mixing is a multiplicative noise process. It follows from eq 3 that a stirring effect appears only if f(x) is a nonlinear function. If f(x) is a linear function, corresponding to a first-order reaction, the noise process becomes additive and stirring effects are absent. The analysis of eq 3 continues by introducing the probability density distribution P(x,t|x0,t0), whose value gives the conditional probability density of finding the system at a point x at time t, provided that it was at x0 at time t0. We use the shorthand notation P(x,t) for this function. The evolution of this probability density is governed by the Fokker-Plank equation
∂tP(x,t) ) -∂x(f(x) P(x,t)) + D∂xx(f′(x)2 P(x,t))
(5)
where the Ito interpretation of eq 3 is used. The stationary solution of eq 5 is given by
[
Ps(x) ) N exp
]
1 x f(z) - Df′(z) f′′(z) dz ∫ D f′(z)2
(6)
where N is a normalizing constant. For the bistable system, the probability distribution function has two maxima. The state
Stirring Effect on the Belousov-Zhabotinsky Reaction
J. Phys. Chem., Vol. 100, No. 49, 1996 19185
Figure 6. Linear relationship between ∆ and σ2 (a) for SS I, [Br-]0 ) 1 × 10-4 M, and (b) for SS II, [Br-]0 ) 1.5 × 10-3 M. The dependence of the proportionality coefficient k on the bifurcation parameter for (c) the SS I and (d) the SS II.
of the system always corresponds to one of these maxima if the intensity of noise is sufficiently small and the gap between steady states is sufficiently large. The experimental results, Figure 4, confirm this behavior of the probability distribution function. The extrema of Ps(x) correspond to the stochastic states. They may be easily found from the equation
f(x) - Df′(x) f′′(x) ) 0
(7)
which follows from the condition dPs(x)/dx ) 0. The roots of eq 7 define the position of stochastic steady states xs. The deterministic steady states xd follow from f(x) ) 0. Our final step is to obtain a relationship between the shift ∆ of the stochastic steady state and the intensity D of noise
x s ) xd + ∆
(8)
We assume that this shift is sufficiently small,
∆ , xd
(9)
This condition is fulfilled for the experimental data, Figures 1 and 2. Substituting eq 8 into eq 7 and performing a Taylor expansion of f(x) results in
∆ ) Df′′(xd)
(10)
According to eq 4, the noise intensity D is proportional to σ2. Its value is measured from the experimental time series and is presented in Figure 2. Therefore, eq 10 gives the following relation between the observed shift ∆ of the stochastic steady state and σ2:
∆ ) kσ2
(11)
∆ ≈ f′′(xd)σ2
(12)
where k ) τf′′(xd), and
where f′′ is the curvature of the rate function. The shift ∆ of the stochastic steady state and the standard deviation σ2 of the
fluctuating signal are related linearly. Since the values of τ and σ2 are positive, the direction of the shift is determined by the sign of f′′(x). Figure 6a,b confirms experimentally the linear relationship between ∆ and σ2 for both steady states in the BZ reaction. Figure 6c,d gives the dependence of the proportionality constant k on the control parameter. The value of k increases on both branches as the control parameter approaches its critical values. 5. Discussion Our analysis deals with the simplest case of stirring effect in chemical bistability where the system behaves effectively onedimensionally. Fluctuations always grow with decreasing stirring rate. The analysis shows that this growth of fluctuations leads to a downward shift of the upper branch and to an upward shift of the lower branch. These shifts depend on the intensity of fluctuations and on the curvature f′′(x) of the rate function. Equations 10 and 12 give the correct direction for the shift of both branches. In the generic case the rate function f(x) has a plot as shown in Figure 5. Its outermost intersections with the x-axis define the deterministic steady states SS I and SS II. The second derivative f′′(x) gives the curvature of the rate function f(x) at these two points, which may be estimated at their deterministic values. This curvature is always negative for SS I and positive for SS II. Therefore, according to eq 10, SS I shifts down and SS II shifts up as the intensity of noise increases. The stochastic steady states are obtained as the intersections of the new function defined by eq 7 with the x-axis. These considerations predict the correct direction of shift of the critical points. To illustrate this, we assume for simplicity that an increase of the control parameter shifts the function f(x) up. This is equivalent to moving the x-axis down and keeping the function f(x) fixed. At a sufficiently low value of the control parameter, f(x) intersects the x-axis only once at SS II. Moving the x-axis down, we arrive at the bistable region. The second intersection with the deterministic S-shaped curve appears at a lower value of the control parameter than for the stochastic curve. Therefore, the critical point CI shifts to the left on the response diagram. The same consideration shows
19186 J. Phys. Chem., Vol. 100, No. 49, 1996 that CII shifts to the right. So, eq 10 gives qualitatively correctly the direction of the shifts of steady states and critical points: briefly, the hysteresis area shrinks along both coordinates as the noise intensity grows. This analysis gives also a quantitative tool to check the stirring effect ∆, namely the proportionality between the observed shift ∆ and the fluctuation intensity σ2. The coefficient of proportionality is proportional to the curvature f′′(x) of the rate function. This curvature increases toward the maxima (minima), i.e. toward critical points. Hence k increases as the control parameter approaches the critical point, in accord with the experimental results presented in Figure 6. Consequently, the shift of the steady state increases near the critical points. Figure 1 shows this kind of experimental behavior. We stress that it is impossible to get such quantitative agreement if the flow rate were chosen as the control parameter. To obtain relations as shown in Figure 6 requires that the control parameter does not affect the hydrodynamics in a CSTR; that is, the hydrodynamics depends on the stirring rate only. Varying the flow rate changes the entire picture of mixing of inflowing reagents, and the dependence of the hysteresis on stirring rate and on control parameter becomes more complicated. Equation 10 illustrates the fluctuation-dissipation theorem according to which the linear system response is given by the autocorrelation function of the fluctuations.21 In particular, our result is similar to that obtained by Hannon and Horsthemke11 for the model of the arsenate-iodate reaction, where a realistic source of fluctuations was taken into account. Their analysis is based on the integro-differential equation of the FokkerPlanck type for the probability distribution function and a concrete choice of the chemical rate function. In contrast, our analysis makes no assumption about the source of fluctuations in a CSTR and about the rate function f(x,µ). To obtain the correct description of these fluctuations a priori, one should use a concrete model of the mixing process. Our experimental data show that such a model should give σ2 ≈ τmix ≈ 1/S. To our knowledge, this analysis predicts qualitatively most of the experimental data concerning the stirring effect in bistable chemical systems. It matches the shifts of critical points and steady states for the clorite-iodide reaction,2-6 the BZ reaction with gallic acid,13 and the experiments reported here. We will show elsewhere that in the iodate-arsenous acid reaction for which a one-dimensional model is known,22 eq 7 predicts quantitatively the S-dependence of the stochastic steady state. The only exception is the minimal bromate oscillator (MBO), for which the hysteresis area broadens and shifts to the right, and the upper steady state shifts up (stabilizes) at low stirring.14,15
Strizhak and Menzinger The simplicity of our analysis is based on the assumption eq 1 that the bistability may be represented by one effective variable. Real chemical systems are often higher dimensional. The increase of the effective dimension may lead to a complicated response to stirring. This shows that the “unusual” stirring effect for the MBO should be explained, at least, in terms of a two-dimensional system. Equation 10 shows that the direction of the hysteresis shift is governed by the curvature of the f(x) in the 1-D case. In the N-dimensional case it should be replaced by the curvatures of the N-dimensional hypersphere, which may be either positive or negative in different directions. As a result, the hysteresis area may shift a priori in any direction when it is projected into the 1-D response diagram. Finally, the analysis performed in this article suggests a new way of utilizing the stirring effect on chemical bistability as a taxonomic tool. If the hysteresis shrinks in response to decreased stirring and both branches are stabilized (stirring effect of the first type), the system may be considered as effectively one-dimensional. Otherwise (stirring effect of the second type), the appropriate kinetic model should be at least two-dimensional. Acknowledgment. This work is supported by the NSERC of Canada. References and Notes (1) Epstein, I. R. Nature 1995, 374, 321. (2) Menzinger, M.; Boukalouch, M.; DeKepper, P.; Boissonade, J.; Roux, J. C.; Saadaoui, H. J. Phys. Chem. 1986, 90, 313. (3) Roux, J. C.; De Kepper, P.; Boissonade, J. J. Phys. 1983, 97A, 168. (4) Luo, Y.; Epstein, I. R. J. Phys. Chem. 1986, 86, 5733. (5) Menzinger, M.; Dutt, A. K. J. Phys. Chem. 1990, 94, 4510. (6) Ochiai, E.-I.; Menzinger, M. J. Phys. Chem. 1990, 94, 8866. (7) Kumpinsky, E.; Epstein, I. R. J. Chem. Phys. 1985, 82, 53. (8) Boissonade, J.; DeKepper, P. J. Chem. Phys. 1987, 87, 2010. (9) Puhl, A.; Nicolis, G. J. Chem. Phys. 1987, 87, 1070. (10) Horsthemke, W.; Hannon, L. J. Chem. Phys. 1984, 81, 4363. (11) Hannon, L.; Horsthemke, W. J. Chem. Phys. 1987, 86, 140. (12) Fox, R. O.; Villermaux, J. Chem. Eng. Sci. 1990, 45, 2857. (13) Dutt, A. K.; Muller, S. C. J. Phys. Chem. 1993, 97, 10059. (14) Dutt, A. K.; Menzinger, M. J. Phys. Chem. 1990, 94, 4867. (15) Dutt, A. K.; Menzinger, M. J. Phys. Chem. 1991, 95, 3429. (16) DeKepper, P.; Boissonade, J. J. Chem. Phys. 1981, 75, 189. (17) Curl, R. L. AIChE J. 1963, 9, 175. (18) Celarier, E.; Kapral, R. J. Chem. Phys. 1987, 86, 3366. (19) Horsthemke, W.; Lefever, R. Noise-Induced Transitions; Springer Verlag: Berlin, 1994. (20) Gardiner, C. W. Handbook of Stochastic Methods, 2nd ed.; Springer: Berlin, 1985. (21) Stratonovich, R. L. Nonlinear Nonequilibrium Thermodynamics I: Linear and Nonlinear Fluctuation-Dissipation Theorems; Springer: Berlin, New York, 1992. (22) Ganapathisubramanian, N.; Showalter, K. J. Phys. Chem. 1980, 80, 4177.
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